Question: All of the 3rd grade teachers and students from Oak went on a field trip to an art museum. Tickets were $$8.50$ each for teachers and $$3.50$ each for students, and the group paid $$53.50$ in total. The next month, the same group visited a science museum where the tickets cost $$34.00$ each for teachers and $$11.00$ each for students, and the group paid $$190.00$ in total. Find the number of teachers and students on the field trips.
Let $x$ equal the number of teachers and $y$ equal the number of students. The system of equations is: ${8.5x+3.5y = 53.5}$ ${34x+11y = 190}$ Solve for $x$ and $y$ using elimination. Multiply the top equation by $-4$ ${-34x-14y = -214}$ ${34x+11y = 190}$ Add the top and bottom equations together. $ -3y = -24 $ $ y = \dfrac{-24}{-3}$ ${y = 8}$ Now that you know ${y = 8}$ , plug it back into $ {8.5x+3.5y = 53.5}$ to find $x$ ${8.5x + 3.5}{(8)}{= 53.5}$ $8.5x+28 = 53.5$ $8.5x = 25.5$ $x = \dfrac{25.5}{8.5}$ ${x = 3}$ You can also plug ${y = 8}$ into $ {34x+11y = 190}$ and get the same answer for $x$ ${34x + 11}{(8)}{= 190}$ ${x = 3}$ There were $3$ teachers and $8$ students on the field trips.